00:01
Okay, here we are given a region, i mean a bounded region enclosed by y being 0 .5 times it to power x square.
00:11
And this region is given in this way.
00:15
X goes from 0 to 1.
00:21
Okay, and this region is enclosed by this part with some, sorry, this blue curve and some horizontal curve.
00:40
Curve, okay, and this is our regions.
00:45
This curve is given as y being this part, one half times it to power x squared.
00:53
Given this region, we want to find the centroid y bar in this shadowed region.
01:02
Okay, now by the formula of the centroid when a y bar is given as one over capital a.
01:11
The capital a is the region for this shadow part times integral, okay, the integral goes from 0 to 1 for the function.
01:30
The integrant or the function in the integral is determined by those two curves, i mean the upper curve, which is the other part minus the lower curve.
01:45
Okay, for convenient that's called as y1 and call the second term as y2.
01:52
We know the formula for the central it is given as 1 of capital n, a capital a times the integral of 4 y1 square minus y2 squared times over 2.
02:05
Okay, this is a formula.
02:09
Okay, use this formula, we know y1 is actually equal to 1 half, each the power of 1 square, which is just equal to 1.
02:19
Okay, y2 is equal to this expression.
02:23
I mean, y2, varies according to different x.
02:27
So plug those information into our integral.
02:30
We have 1 over capital a, 0 to 1, 1 half times 1 half is the power 1 to the power 2 minus 1 half is the power x squared to the power to dx.
02:49
Okay, then off some simplification.
02:52
We know this is this can be written as 1 over 2a.
02:57
Integral from 0 to 1, e squared over 4, minus each to the power of 2x squared over 4 dx.
03:10
For this term, as it is a constant, so use the linearity of the integral, this is equal to 1 over 2a times just e squared over 4, minus 1 over 4, the integral from 0 to 1, each the power 2x squared d x.
03:30
Okay, for the second term, as we don't have a very explicit expression for the anti -derivity for each the power 2x squared.
03:42
So we need to use the so -called simpson's rule.
03:45
I mean, use the simpsons rule, you know, the integral from a to b, fx, dx, is approximately equal to b minus a over 6 times f of a plus f of b plus no simple.
04:03
Important term four times f of the middle point between a and b.
04:12
Okay, here, regardless of this as our f, we know f of 0 is equal to each the power 0, which is equal to 1.
04:21
F of 1 is equal to each the power 2.
04:26
Now, what about a plus b over 2? 1 plus 0 over 2 is just equal to 1 half.
04:32
So f of one half is equal to equal to equal to power 2 times 1 over 4, which is equal to each the power 1 over 2...